There are inexpensive little porcelain graters on the market designed specifically for ginger. They have little nubs that catch on the ginger’s fibers and shred it, and it is easy to remove the ginger from the grater to add to your dish. There are also very fine metal graters available that, likewise, are meant especially for ginger. If you can squeeze another gizmo in your kitchen, you might prefer one of these to your generic grater, which is a little coarse for ginger.
You will also find the ginger easier to grate if you freeze it first.